(1) First, one extends the boundary conditions of [1]} on the fermionic fields along the lines of [2]}, by adding to the standard \(\mathcal {O}(r^{-2})\)  behaviour a gradient term that decays slowlier at infinity, \(\psi _{k} = \partial _k \chi + \mu ^{\prime }_k\)  with \( \chi = \mathcal {O}\left(1\right)\)  and \(\mu ^{\prime }_{k} =\mathcal {O}(r^{-2})\) . The term \(\partial _k \chi \)  is the leading piece of a supersymmetry transformation of the gravitino field \(\delta _\chi \psi _k\) . The subsequent terms in \(\delta _\chi \psi _k\)  are of lower order and can be included in \(\mu ^{\prime }_k\) . The same is true for the supersymmetry variations of the bosonic fields, which are of the same type as the existing terms \(\overline{h}_{ij}(\theta , \varphi )/r\)  and \(\overline{\pi }^{ij}(\theta , \varphi )/r^{2}\) , as it will indirectly follow from our subsequent analysis. The idea of the new extended boundary conditions, thus, is that the supergravity fields are defined at spatial infinity up to a supersymmetry transformation. Because this supersymmetry transformation has a non-vanishing charge, it is an improper gauge transformation with non trivial physical content, and it cannot be gauged away [3]}. Once included, it must be kept.
